A spherical tank is 3 m tall and therefore has a capacity of about 14000 L (NOTE that 3m is its diameter, and one cubic meter is 1000 L). You have been asked to put level lines on the tank showing the depths where the tank contains 1000, 5000, and 10000 L of liquid. (a) Use calculus to determine the height of each line (to the nearest cm). (b) Plot the circular cross-section of the tank and the level lines (NOTE the lines do not have to stay within the circle).

(a) 1) When V = 1000 L, the height of the liquid in the tank is approximately 62 cm

2) When V = 5,000 L, the height of the liquid in the tank is approximately 106 cm

3) When V = 10,000 L, the height of the liquid in the tank is approximately 202 cm

b) The drawing of the spherical tank showing the height of the liquid level for each volume created with Microsoft Excel is attached

Step-by-step explanation:

(a) The height of the spherical tank = 3 m

The diameter of the spherical tank, d = 3 m

The capacity of the spherical tank = 14,000 L

The volume of a sphere, V = (4/3) × π × r³ =

Where, r = d/2

∴ r = 3 m/2 = 1.5 m

V = (4/3) × π × (1.5)³ ≈ 14.1371669412 m³

Using calculus, we have;

For the two halves, we have;

1) For V = 1000 L, we have;

1000 L =

When V = 1000 L, we have;

The liquid level, y ≈ 0.62 m = 62 cm

2) When V = 5,000 L, we have;

The liquid level, y ≈ 1.06 m = 106 cm

3) When V = 10,000 L, we use the height of the unfilled volume at the top which is the volume of the sphere less the 10,000 L as follows;

For the volume of the sphere of 14,000 L, we have;

Volume at the top = 14,000 L - 10,000 L = 4,000 L = 4 m³

Therefore

The height of the liquid in the tank, y = d -

∴ y = 3 m - 0.985 m = 2.015 m

The liquid level, y ≈ 2.015 m = 201.5 cm ≈ 202 cm

b) Please find attached the drawing of the spherical tank created with Microsoft Excel

in problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion. to find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

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step-by-step explanation: